The shell model for the exchange-correlation hole in the strong-correlation limit

Abstract

We present a model for the exchange-correlation hole and the exchange-correlation energy in the strong-correlation (SC) limit of density functional theory. The SC limit is useful in the construction of exchange-correlation functionals through interpolation of the adiabatic connection. The new approximation (referred to as shell model) is an improvement of the non-local radius (NLR) model recently proposed by Wagner and Gori-Giorgi [Phys. Rev. A 90, 052512 (2014)]. The NLR model does not correctly reproduce the limit of the strongly correlated homogeneous electron gas and this shortcoming is remedied by the shell model. As in the case of the NLR model, the spherically averaged electron density ρ(r,u)=∫dΩ4πρ(r+u) is the starting point for the construction of the shell model and it is also its computational bottleneck. We show how ρ(r, u), the NLR, and the shell model can be implemented efficiently. For this purpose, analytical integrals for the normalization and the energy density of the underlying holes are provided. Employing the shell model, we illustrate how improved adiabatic connection interpolations can be constructed.

Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy

“…22 Despite these appealing features, there are two main obstacles to the routine use of the SCE functional: its availability is restricted to small systems 16,23 and its energies are way too low for most of physical and chemical systems. 19,23−25 The nonlocal radius (NLR) functional, 26 and the newer shell model 27 are inspired to the SCE functional form and retain only some of its nonlocality. They are readily available, 27 but, being approximations to the SCE functional, their energies are also too low with respect to those of chemical systems.…”

“…They are readily available, 27 but, being approximations to the SCE functional, their energies are also too low with respect to those of chemical systems. 26,27 The information encoded in the SCE functional or its approximations can be combined with the complementary information from the weak coupling limit. This has been recently used for constructing XC functionals from a local interpolation along the adiabatic connection.…”

“…This has been recently used for constructing XC functionals from a local interpolation along the adiabatic connection. 14,27−29 Although this approach is promising for treating strong correlation within the realm of DFT, 14 it can still easily overcorrelate (for example for stretched bonds it overcorrelates the fragments), again because the SCE (exact or approximate) quantities are often far from the physical ones. 29 …”

“…Motivated by the structure of eq 5, and similarly to the recent nonlocal approximations for the SCE functional for three-dimensional systems, 26,27 we introduce the spherically averaged density ρ̃( r , u ) around a position r ,and the function N e ( r , u ),These functions have been studied and efficiently implemented by Ernzerhof and co-workers. 27,28,51 We now want to find a physical approximation for the crucial quantitieswhich give the expected number of electrons in a sphere of radius R i λ ( r ) centered at the reference electron in r . We notice at this point that in an inhomogeneous system, even one-dimensional, it is not possible to write the exact SCE radii explicitly in terms of the function N e ( r , u ) obtained by spherically averaging the density around a reference electron in r as in eq 8.…”

“…Although the functional is highly nonlocal, it can be obtained at a computational cost comparable to that of the NLR and shell functionals, which have been recently implemented in a very efficient way. 27 Many strategies to improve the accuracy can be pursued: the inclusion of kinetic correlation through interpolation along the adiabatic connection (as in Figure 4); trying to model directly the λ-dependence of σ i ; the generalization to noninteger number of electrons 22 and spin densities; improving the accuracy for the UEG, and adding the dependence on the gradient of S i ( r ). The functional can also be readily applied to other dimensionalities, e.g., electrons confined in quasi-1D and quasi-2D geometries, for which the SCE approach has already proven very useful.…”